[seqfan] Eigen-sequences from generic beanstalk-theme?

[Antti Karttunen]

Let sequence s be strictly growing, with s(1) = 1.

Let sequence b(n) = number of terms of sequence s needed for a sum
s(i)+s(j)+...+s(k) = n, when selected by a greedy algorithm starting
from the largest s(i) <= n, and always choosing the largest term of s
that still fits into the sum. This might be equal to the minimum
number of terms of s needed to represent n, depending on how the
sequence s turns out to be.

Let sequence c(n) = number of iterations of map (k - b(k)) --> k
needed to reach s(n)-1 when starting from k = s(n+1)-1. [This might
turn out the same as number of such steps to reach s(n) from s(n+1).]

Are there any interesting solutions such that we would have s(n) =
c(n+d) after some n > 0, where d is a small shifting constant, like 0
or 1 ?

If too many solutions, then add extra conditions for sequence b, like
that the terms of s to be added up must be distinct or
non-consecutive, etc.


Best,

Antti